Explicit Closed-form Solution Research Articles

Steady state analysis for discrete tracking filters

An analysis of a discrete constant-gain alpha - beta - gamma tracking filter is presented. Steady-state tracking filter gains are determined without solving a discrete matrix Ricatti equation. Explicit closed-form gain solutions are found as a function of the roots to a cubic equation. Gain variations as a function of process noise, measurement noise, and sampling time are investigated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Symbolic eigenvalue analysis for adaptive stepsize control in PNS shock stabilization

Abstract Parabolized Navier-Stokes (PNS) solution techniques for high speed compressible flow involve marching in a “time like” spatial coordinate but have been observed in practice to be sensitive to flow conditions and require stabilization by numerical dissipation techniques. In this study we develop an eigenvalue analysis for shock stability. Using the algebraic symbolic manipulator MACSYMA, an explicit closed-form solution for the shock system eigenvalues is developed. This is then applied on the evolving shock contour to determine the stability limit on the PNS stepsize. Numerical experiments confirm the validity of the approach and demonstrate that the approach can be implemented explicitly to yield an adaptive stepsize algorithm and thereby a more robust PNS scheme.

Aggregation of Information in Large Cournot Markets

Consider a homogeneous product market where firms have private information about an uncertain demand parameter and compete in quantities. We examine the convergence properties of Bayesian-Cournot equilibria as the economy is replicated and conclude that large Cournot (or almost competitive) markets do not aggregate information efficiently except possibly when the production technology exhibits constant returns to scale. Even in a competitive market there is a welfare loss with respect to the first best outcome due to incomplete information in general. Nevertheless a competitive market is efficient, taking as given the decentralized private information structure of the economy. Endogenous (and costly) information acquisition is examined and seen to imply that the market outcome always falls short of the first best level with decreasing returns to scale. The results are also shown to be robust to the addition of extra rounds of competition which allows firms to use the information revealed by past prices. Explicit closed form solutions yielding comparative static results are obtained for models characterized by quadratic payoffs and affine information structures.

Analytical solutions for oblique wave growth from a ring-beam distribution

Analytical solutions are presented for the linear growth rate of oblique plasma waves in a magnetized plasma due to resonant interactions with a model ringbeam distribution. Explicit closed-form solutions for the angular dependence are obtained in terms of modified Bessel functions of the first kind. In the limits of either quasi-longitudinal or quasi-transverse propagation the analytical solutions take the form of simple algebraic expansions, which allow an immediate comparison of the relative contributions from different harmonic resonances, and which also determine the conditions for marginal stability for any specific resonance. The results can be applied, for instance, to the growth of waves following ionization of neutrals originating from cometary, planetary, or interstellar material in the solar wind. In a weakly unstable plasma the analytical results also provide an important check on the complex numerical codes that hitherto constituted the only method available for evaluating the growth of oblique plasma waves.

The ellipsoidal cavity in the presence of a low-frequency elastic wave

We consider the problem of scattering of a longitudinal or a transverse plane elastic wave by a general ellipsoidal cavity in the low-frequency region. Explicit closed-form solutions for the zeroth- and first-order approximations are provided in terms of the physical and geometric characteristics of the scatterer, as well as the direction cosines of the incidence and observation points. This was made possible with the introduction of an analytical technique based on the Papkovich representations and their interdependence. The leading low-frequency term for the normalized spherical scattering amplitudes and the scattering cross section are also given explicitly. Degenerate ellipsoids corresponding to the prolate and oblate spheroids, the sphere, the needle, and the disc are considered as special cases.

Matrix methods in discrete-time quantum mechanics.

The operator difference equations that arise in the finite-element treatment of a quantum theory are implicit and therefore difficult to solve. By introducing a matrix formulation it is possible to circumvent the implicit character of these equations and obtain explicit closed-form solutions for arbitrary matrix elements of any operator.

The finite deformation of a pressurized circular tube for a class of compressible materials

This investigation is concerned with the problem of a hollow circular cylinder subjected to uniform internal and external pressure within the equilibrium theory of finite elasticity. The tube is composed of homogeneous, isotropic,compressible materials of special type, namely harmonic materials. Explicit closed-form solutions for the deformation and stress fields are obtained. The true stress distribution, expressed as a function of the undeformed coordinates, is shown to be essentially independent of material properties. The two cases of internal pressure only, and external pressure only, are examined in detail. In the former case, there is a critical value of the applied pressure at which the maximum hoop stress in the tube, occurring at the inner surface, becomes unbounded. Results appropriate for thin shells are also obtained. For the case of external pressure only, a critical value of the applied pressure exists for which closing of the cavity is predicted. For nearly solid cylinders, or equivalently, for a cavity in an unbounded medium, explicit results are provided for the corresponding stress concentration factor.

The pressurized hollow sphere problem in finite elastostatics for a class of compressible materials

This investigation is concerned with the problem of a hollow sphere subjected to uniform internal and external pressure within the equilibrium theory of finite elasticity. The sphere is composed of homogeneous, isotropic, compressible materials of special type, namely harmonic materials. Explicit closed-form solutions for the deformation and stress fields are obtained. The true stress distribution, expressed as a function of the undeformed coordinates, is shown to be essentially independent of material properties. The two cases of internal pressure only, and external pressure only, are examined in detail. In the former case, there is a critical value of the applied pressure at which the maximum hoop stress in the sphere, occurring at the inner surface, becomes unbounded. Results appropriate for thin shells are also obtained. For the case of external pressure only, a critical value of the applied pressure exists for which the cavity closes. The maximum hoop stress does not always occur at the cavity wall. For nearly solid spheres, or equivalently, for a cavity in an unbounded medium, explicit results are provided for the corresponding stress concentration factor. For sufficiently small values of applied pressures, all the foregoing results coincide with those of classical linear isotropic elastostatics.

An explicit solution to an optimal tax problem

Under specific but perhaps not over-restrictive assumptions on social welfare and consumer preferences, an explicit closed-form solution for an optimal linear income tax is derived. Specifically, given linear income supply functions and a rank-order social welfare function, the optimal tax rate and benefit level are characterized by four parameters: I, a measure of pre-tax inequality in the ability (wage) distribution; r, the fraction of potential total income required for (non-redistributed) government revenue; σ, the fraction of potential total income required for consumer subsistence expenditures; and a disincentive parameter, δ, the marginal propensity to spend on leisure or the amount by which earned income is reduced in response to a unit increase in unearned benefit. Defining ∅, the ratio I/(1 - σ - r), the optimal tax rate τ is given by: τ= 1 1−δ − δ 2φ(1−δ) 2 1+ 4(1,minus;δ)φ δ 1 2 −1 The formula is used to fully characterize τ in terms of the parameters. Results include the following: τ = 0 if I = 0; τ = 1 if δ = 0; τ is increasing in I, σ and r; τ may be increasing or decreasing in δ depending on the value of ∅; when disincentive effects are large, τ becomes close to ∅ so that, in such economies, if σ and r are small, the optimal tax rate is equal to the measure of pre-tax inequality. Formulae for the deadweight loss associated with the tax are derived and some observations are offered on the empirical issues associated with the model.

Optimal Smoothing of Empirical Bode Plots by Hilbert Transform Reciprocation

Measurement errors corrupting the data of an empirical Bode plot of a causal system (log-amplitude and phase, versus frequency) can be minimized by taking into account the theoretical redundancy of such data in the ideal (noise-free) case, wherein each plot is the Hilbert transform of the other. An explicit closed-form analytical solution of the problem of least-mean-square fit of reciprocal Hibert-transform pairs to the measured data is derived, and illustrative examples are presented.

Explicit solution of the mean spherical model for ions and dipoles

It is shown that the solution of the mean spherical approximation for the ion-dipole mixtures obtained by Blum, Adelman, and Deutch has an explicit closed form solution which is one of the roots of a cubic equation.

Resolution of a Core Problem in Wound Rolls

Previous misunderstandings about the behavior of stresses in the vicinity of the core of wound rolls are resolved through the development of an explicit closed-form analytical solution for the radial and circumferential stresses which are generated during winding; the solution holds in the general case of variable winding tension. Asymptotic series are derived and then employed to compute profiles of stresses within the wound material for several cases of practical interest. A parametric analysis of the influence of core elasticity on structural integrity of the roll underscores the indispensable support provided by the core at the inner boundary of the roll. Results show that the circumferential stress in the vicinity of the core strongly depends on core stiffness. A relation that is derived between interlayer pressure and circumferential stress at the core boundary may be useful as a guide in core design and in preventing roll collapse.

Geometric methods for nonlinear optimal control problems

It is the purpose of this paper to develop and present new approaches to optimal control problems for which the state evolution equation is nonlinear. For bilinear systems in which the evolution equation is right invariant, it is possible to use ideas from differential geometry and Lie theory to obtain explicit closed-form solutions.

Analysis of periodically switched linear circuits

A new approach, to the exact analysis of linear circuits containing periodically operated switches is presented. After reformulation of the state equations conventional Fourier analysis is used to determine the response to arbitrary deterministic or stochastic inputs. The analysis is applicable also to improper circuits and circuits causing discontinuities in the state variables at the switching instants. The switches may operate in an arbitrary fashion with a common switching period. Explicit closed form solutions for both the (Fourier coefficients of the) periodically time-varying transfer function \hat{H}(f, t) and the impulse response h(t,\tau) are derived. These results are most suitable for computer aided design. Applications to switched filters and modulators are given.

Free boundaries for plasmas in surface magnetic field configurations

Exact analytic solutions for the plasma/magnetic-field free-boundary problem are derived for three planar, surface magnetic field configurations: the picket fence, the simple magnetic slab, and the staggered magnetic slab. In the ideal magnetohydrodynamic limit, conformal mapping techniques yield explicit closed-form solutions for the free boundaries, the magnetic field lines, and the magnetohydrodynamic stability criteria. For configurations with plasma confined to one side of a continuous, sharp free boundary, it is found that with a given choice of the physical parameters (i.e., dimensions, current, and plasma pressure), there may be only one stable solution; only one unstable solution; two solutions, one stable and one unstable; or no solution at all. For the staggered slab, parameters can be chosen so that the conductors nearest the plasma are force free, an arrangement which could minimize the size of physical supports and the associated plasma loss in confinement applications.

On the voltage distribution across a string of suspension insulators

A difference equation formulation is given for the problem of determining the voltage distribution across a string of suspension insulators, taking into account the capacitance Cg between the insulator hardware and the guard ring, and the conductance of the insulator. Explicit closed-form solution is given for the case in which Cg does not vary from unit to unit and the form of tapering needed to make a uniform distribution of the voltage is worked out.

Exact Analysis of Linear Circuits Containing Periodically Operated Switches with Applications

An exact analysis of linear circuits containing periodically operated switches with exponentially modulated cisoidal inputs using the state-space approach is presented. Explicit closed form solutions for both the time and frequency domain suitable for digital computation are obtained. The analysis is applicable to any circuit configuration and can handle cases in which discontinuities in state variables occur at the switching instants. Applications of this analysis to the design of switched filters and modulators are discussed.

Optimal signal design for sequential signaling over a channel with feedback

Turin [1] has studied the problem of optimal sequential detection over a channel with feedback when the signals used are subject to peak and average power constraints. With special additional constraints on the signals he was able to obtain a series solution for the first passage probability density and hence find optimal signals when the ratio of peak power to average power was infinity or one. Horstein,[2] concurrently with this study, was able to find optimal signals for arbitrary ratios of peak to average power when the signals were subject to the same special constraints. This paper presents a simpler and more direct approach for obtaining optimal signals which minimize the expected time to decision for arbitrary peak power constraints and various average power constraints. Explicit closed form solutions are derived for the expected energies and times to decision without having to first obtain the probability distribution of the first passage time. No special constraints are required on the signals. For communication with white noise in the feedback link, optimal stationary signals are found for peak and average power constraints. It is shown that communication rates up to but strictly less than channel capacity are possible with noisy feedback. However, it is also noted that for the schemes considered here, the stationary signals and white noise in the feedback link imply infinite feedback power.